World Map



World Map with Confirmed and Death Cases (Data source (JHU 2020b), see also (JHU 2020a))

Total Confirmed for each Country

Total Deaths for each Country

Bar Chart



Bar Charts with descending order

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Bar Chart Confirmed

Bar Chart Table Deaths

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Table Confirmed and Deaths - Cumulated and Daily Cases

Case numbers are taken from (JHU 2020b)

Bar Chart / Inhabitants



Bar Charts with descending order

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Bar Chart Confirmed per 100,000 Inhabitants

Bar Chart Deaths per 100,000 Inhabitants

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Table Confirmed and Deaths - Cumulated and Daily Cases per 100,000 Inhabitants

Population numbers are taken from (CIA 2020a), (CIA 2020b)

Bar Chart - CFR



Bar Chart with descending order

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Bar Chart CFR Total - Case Fatality Rate (in %)

Bar Chart CFR Total (in %) - - selected countries

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Case Fatality Rate - Proportion of deaths from confirmed cases

The number of confirmed cases is an early predictor of the number of deaths. The number of today’s deaths is already determined by the infections about by \(\sim19\) days ago or respectively by the confirmed cases about by \(\sim11\) days ago. The Confirmed infection to Death period can be assumed to be an average of \(11\) days (see RWI 2020).

However, this varies considerably depending on country-specific test rate and health system. In the worst health systems it is only one day for recognized cases, the “Confirmed” cases must be “lagged” by \(\sim1\) day. In the best case, the time from the end of incubation period (in average \(\sim5-6\) days) to death is an average \(\sim19\) days. In this case, the average Confirmed infection to Death period is \(\sim14\) days), the “Confirmed” cases must be correctly “lagged” by \(\sim14\) days. For the assumed time periods see (RKI 2020c), (RKI 2020b), for Case Fatality Rate and Incubation Period in general see (Wikipedia contributors 2020a), (Wikipedia contributors 2020c).

The simple calculation with unlagged cumulated confirmed cases divided by cumulated deaths results in a significant underestimation of the CFR in health systems with early disease detection.

The Infection Fatality Rate (IFR) is the fatality rate of all infection, that means detected confirmed cases and undetected cases (asymptomatic and not tested group). This lethality is assumed to be country independent and only rough estimates exist (RKI: bottom of existing estimates \(\sim0.56\%\)).

Case Fatality Rate in % - CFR_total and CFR_past_period (period of past 14 days) w/ Confirmed lagged by 11 days

Case Fatality Rate (in %) - selected countries

Country CFR_total CFR_past_period CFR_unlagged
Austria 3.5 4.1 3.2
France 23.2 18.4 13.0
Germany 4.1 4.2 3.3
India 8.8 7.6 3.2
Italy 16.8 12.2 13.3
South Korea 2.3 4.0 2.2
Spain 13.6 7.8 10.4
United States of America 9.1 8.1 5.4

Cumulated and Daily Trend



Cumulated and Daily Cases over Time

Row

World

Row

Selected Countries

China

Austria

France

Germany

Italy

India

South Korea

Spain

United States of America

Germany - Confirmed and Deaths

Virus Spread on log10 scale



Column

Exponential Growth Evaluation

China and South Korea have slowed down significantly the exponential growth. Therefore, their lines in the chart with the log10 scale have no longer a significant slope.

In early phases countries have a more or less unchecked exponential growth. If countermeasures are effective, reduced exponential growth is reflected in a reduced slope of the accumulated cases again.

Column

Virus Spread with log10 scale (since mid of Jan)

Column

Virus Spread / Inhabitants with log10 scale (since mid of Jan)

Exponential Growth



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Estimation spread of the Coronavirus with Linear Regression of log data

Exponential Growth and Doubling Time \(T\)

Exponential growth over time can be fitted by linear regression if the logarithms of the case numbers is taken. Generally, exponential growth corresponds to linearly growth over time for the log (to any base) data (see also Wikipedia contributors 2020b).

The semi-logorithmic plot with base-10 log scale for the Y axis shows functions following an exponential law \(y(t) = y_0 * a^{t/\tau}\) as straight lines. The time constant \(\tau\) describes the time required for y to increase by one factor of \(a\).

If e.g. the confirmed or death cases are growing in \(t-days\) by a factor of \(10\) the doubling time \(T \widehat{=} \tau\) can be calculated with \(a \widehat{=} 2\) by

\(T[days] = \frac {t[days] * log_{10}(2)} {log_{10}(y(t))-log_{10}(y_0)}\)

with

\(log_{10}(y(t))-log_{10}(y_0) = = log_{10}(y(t))/y_0) = log_{10}(10*y_0/y_0) = 1\)

and doubling time

\(T[days] = t[days] * log_{10}(2) \approx t[days] * 0.30\).

For Spain, Italy, Germany we have had a doubling time of only \(T \approx 9days * 0.3 \approx 2.7 days\) !!.

The doubling time \(T\) and the Forecast is calculated for following selected countries: Austria, France, Germany, Italy, India, South Korea, Spain, United States of America and World in total (see Forecast / Doubling Time).

Germany - Trend with Forecast on a linear scale

Forecast Plot - next 14 days

The plot shows the extreme forecast increase in case of unchecked exponentiell growth. The dark shaded regions show 80% rsp. 95% prediction intervals. These prediction intervals are displaying the uncertainty in forecasts based on the linear regression over the past 7 days.

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Comparison Exponential Growth

Germany - Example plot with ~linear slope on a log10 scale

Compare Exp vs Linear Growth



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Comparison Exponential vs. Linear Growth

The charts compare the different forecasts for an exponential rsp. linear growth model. Due to the large fluctuations of the daily cases regression of three weeks is required. Otherwise the prediction levels are much too big.

The dark shaded regions are indicating the \(80\%\) rsp. \(95\%\) prediction intervals. These prediction intervals are displaying the “pure” statistical uncertainty in forecasts based on the regression models.

For doubling periods in the order of period of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty, see (RKI 2020b), we no longer have exponential growth. The “old” infected cases are at the end of the doubling period no longer infectious (active). This results in a constant infection rate with basic reproduction number \(R_0 \sim 1\) or even \(<1\).

Note: for case numbers of German federal states see (RKI 2020a).

Column

Cumulated Cases - Comparison Exponential and Linear Growth

Daily Cases - Comparison Exponential and Linear Growth

Doubling Time / Forecast



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Doubling Time and Forecast The forecasted cases for the next 14 days are calculated ‘only’ from the linear regression of the logarithmic data and are not considering any effects of measures in place. In addition data inaccuracies are not taken into account, especially relevant for the confirmed cases.

Therefore the 14 days forecast is only an indication for the direction of an unchecked exponentiell growth.

Forecast (FC) with linear regression: Doubling Time (days), Forecasted cases tomorrow and Forecasted cases in 14 days
Country Case_Type T_doubling last_day FC_next_day FC_14days
Austria Confirmed 103.3 14’795 14’944 16’306
France Confirmed 24.0 156’480 163’580 238’223
Germany Confirmed 36.8 147’065 150’984 192’938
India Confirmed 8.4 18’539 20’276 59’212
Italy Confirmed 36.7 181’228 185’536 237’151
South Korea Confirmed 404.2 10’674 10’700 10’942
Spain Confirmed 27.5 200’210 207’769 288’354
United States of America Confirmed 16.0 784’326 826’853 1’450’597
World Confirmed 18.4 2’472’258 2’586’924 4’224’649
Austria Deaths 20.1 470 488 763
France Deaths 17.4 20’292 21’528 36’095
Germany Deaths 11.9 4’862 5’271 11’274
India Deaths 9.6 592 643 1’645
Italy Deaths 30.8 24’114 24’770 33’183
South Korea Deaths 70.6 236 239 271
Spain Deaths 30.0 20’852 21’511 29’052
United States of America Deaths 8.3 42’094 48’308 143’766
World Deaths 13.7 169’985 183’094 353’417

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Check of Forecast Accuracy

The forecast accuracy is checked by using the forecast method for the nine days before the past three days (training data). Subsequent forecasting of the past three days enables comparison with the real data of these days (test data).

The comparison is also an early indicator if the exponential growth is declining. However, possible changes in underreporting (in particular the proportion confirmed / actually infected) requires careful interpretation.

For doubling periods in the order of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty, (see RKI 2020b), we no longer have exponential growth. Since the “old” infected cases are no longer infectious after these periods and we then have a constant infection rate with basic reproduction number \(R_0 \sim 1\).

Instead, we have “only” linear growth of the cumulative Confimred Cases and the Daily Confirmed Cases remain more or less constant or even decrease.

However, the basic reproduction number \(R_0\) is a product of the average number of contacts of an infectious person per day, the probability of transmission upon contacts and the average number of days infected people are infectious. With the current uncertainty of the average duration of the infectivity duration, \(R_0\) can therefore be estimated from the doubling time only to a very limited extent. See also (CMMID 2020) and (RKI 2020d).

Germany - Forecast Accuracy for past three days

Forecast



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Forecasting with lagged Predictors

The number of confirmed cases is an early predictor of the number of deaths. The number of today’s deaths is already determined by the infections about by \(\sim19\) days ago or respectively by the confirmed cases about by \(\sim11\) days ago see Bar Chart - CFR.

The country specific case fatality rate (CFR, proportion of deaths from confirmed cases) indicates the country specific testing rate and may depend on quality/capacity of hospitals.

Overall a rough conclusion on the country specific underreporting rate (lack of diagnostic confirmation; proportion of all infected to confirmed cases) is feasible if the infection fatality rate (IFR, confiremd cases plus all asymptomatic and undiagnosed infections) is assumed to be country independent and the IFR is known (bottom of existing estimates \(\sim0.56\%\), assumption by RKI see (RKI 2020b).

In this case an estimation of the CFR of \(0.06\) \((6\%)\) indicates an underreporting by a by a factor of \(\sim10\). A CFR of \(0.20\) \((20\%)\) indicates an underreporting by a by a factor of \(\sim30\). This corresponds to RKI assumption of a underreporting by a factor of \(11-20\) (RKI 2020c). Unfortunately, the IFR or lethality is still far too imprecise for concrete conlusions.

In the model paper RKI assumes for the

  • Incubation period \(\sim5-6\) days - Day of infection day until symptoms are upcoming)
  • Hospitalisation \(+4\) days - Admission to the hospital (if needed) after Incubation Period)
  • Average period to death \(+11\) - if the patient dies, it takes an average of \(11\) days after admission to the hospital

Depending on the country-specific test frequency (late or early tests), the

*lag_days - time from receipt of the confirmed test result to death, Confirmed to Death, is about \(11-13\) days.

Note: these methods are also used for example for advertising campaigns. The campaign impact on sales will be some time beyond the end of the campaign, and sales in one month will depend on the advertising expenditure in each of the past few months (see Hyndman and Athanasopoulos 2020).

Column

Daily Confirmed and Death Cases

Lag days and Case Fatality Rate (CFR)

Case Fatality Rate in % (CFR, proportion of deaths from confirmed cases) lagged by 11 days; total and for period of past 14 days
Country CFR_total CFR_past_period CFR_unlagged
Germany 4.1 4.2 3.3
Italy 16.8 12.2 13.3
Spain 13.6 7.8 10.4

Column

Daily Deaths depending on lagged Daily Confirmed Cases

Exampla Germany - White Noise of Forecast Residuals

Forecast residuals indicate quality of fit with Arima model:

References



Data Source

Data Source

Data files are provided by Johns Hopkins University on GitHub
https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data/csse_covid_19_time_series

  • Data files:
    • time_series_covid19_confirmed_global.csv
    • time_series_covid19_deaths_global
    • time_series_covid19_recovered_global.csv

The data are visualized on their excellent Dashboard
Johns Hopkins University Dashboard
https://coronavirus.jhu.edu/map.html

Code Source

Directory with all R Sources is replicated in GitHub repository:
https://github.com/WoVollmer/R-TimesSeriesAnalysis/tree/master/Corona-Virus

Code is based on ideas from https://rpubs.com/TimoBoll/583802

Bibliography

CIA. 2020a. “The World Factbook - Country Comparison :: Population.” Internet. https://www.cia.gov/library/publications/resources/the-world-factbook/fields/335rank.html.

———. 2020b. “The World Factbook - Population Raw Data.” Internet. https://www.cia.gov/library/publications/resources/the-world-factbook/fields/rawdata_335.txt.

CMMID. 2020. “Temporal Variation in Transmission During the Covid-19 Outbreak.” GitHub. https://cmmid.github.io/topics/covid19/current-patterns-transmission/global-time-varying-transmission.

Hyndman, Rob J, and George Athanasopoulos. 2020. “Forecasting: Principles and Practice.” Online textbook. https://otexts.com/fpp3/lagged-predictors.html.

JHU. 2020a. “Coronavirus - Dashboard.” Internet. https://coronavirus.jhu.edu/map.html.

———. 2020b. “Coronavirus - Time Series Data.” GitHub. https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data/csse_covid_19_time_series.

RKI. 2020a. “Coronavirus - Fallzahlen.” Germany: Internet. https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Fallzahlen.html?nn=13490888.

———. 2020c. “Coronavirus - Steckbrief.” Germany: Internet. https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Steckbrief.html.

———. 2020d. “Schätzung Der Aktuellen Entwicklung Der Sars-Cov-2-Epidemie in Deutschland - Nowcasting.” Germany: Internet. https://www.rki.de/DE/Content/Infekt/EpidBull/Archiv/2020/17/Art_02.html?nn=13490888.

RWI. 2020. “Corona-Pandemie: Statistische Konzepte Und Ihre Grenzen.” Germany: Internet. http://www.rwi-essen.de/unstatistik/101/.

Wikipedia contributors. 2020a. “Case Fatality Rate — Wikipedia, the Free Encyclopedia.” https://en.wikipedia.org/w/index.php?title=Case_fatality_rate.

———. 2020b. “Exponential Growth — Wikipedia, the Free Encyclopedia.” https://en.wikipedia.org/w/index.php?title=Exponential_growth.

———. 2020c. “Incubation Period — Wikipedia, the Free Encyclopedia.” https://en.wikipedia.org/w/index.php?title=Incubation_period.